Numerical differential continuation approach for systems of nonlinear equations with singular Jacobian

Document Type : Original Article

Author

Department of Mathematics, Tafresh University

Abstract

It is well known that, one of the useful and rapid methods for a nonlinear

system of algebraic equations is Newton's method. Newton's method has at least

quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood

of the solution. In this paper, a differential continuation method is presented for

solving the nonlinear system of algebraic equations whose Jacobian matrix is singular

at the solution. For this purpose, at fi rst, an auxiliary equation named the homotopy

equation is constructed. Then, by differentiating from the homotopy equation, a

system of differential equations is replaced instead of the target problem and solved. In

other words, the solution of the nonlinear system of algebraic equations with singular

Jacobian is transformed to the solution of a system of differential equations. Some

numerical tests are presented at the end and the computational efficiency of the

method is described.

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